Supercomputing Institute Fall 2000 Bulletin Vol. 17 No. 2

In modeling complex materials and their interfaces, it is often necessary to take account of the dynamics of a large number of atoms and electrons. Problems of this kind arise, for example, in such diverse fields as magnetism, corrosion science, electrochemistry, polymer science, and biology. One approach to this problem is a direct dynamics calculation in which the equations are solved for the electronic structure at each step in the classical evolution of the positions of the atomic nuclei of a large system.

Supercomputing Institute Fellow Professor J. Woods Halley (Physics and Astronomy) and his research group are developing a version of this approach that is suitable for transition metals, or materials involving them; this approach is computationally tractable for complex systems. The basic idea is to use the same kind of first principles electronic structure calculation (i.e., a plane wave local density approximation calculation) that is used in the Car-Parrinello method to parametrize a simpler version of the electronic structure problem-self-consistent tight binding molecular dynamics; this kind of calculation is not as expensive to solve at each step in an atomic dynamics. The term "self-consistent" refers to Hartree self-consistency, familiar from the well known Hartree and Hartree-Fock approximations. The self-consistent aspect distinguishes this method from the original tight-binding and extended Hückel approaches. Electronic properties of the isolated ions of the constituent atoms of the material being simulated are directly incorporated into the model from experiment and cause the on-site energies of the ions in the model to depend on the total number of electrons on the ion. In this way, the difficult problem of the local electronic structure around an atom does not have to be solved repeatedly while studying many-atom systems.

Patrick Schelling, now at Argonne National Laboratory, developed many aspects of this approach in his thesis while with the Halley group. He applied it to the study of rutile titanium dioxide surface structure and polaron structure in rutile titanium dioxide, to grain boundaries in rutile, and to titanium metal-titanium dioxide interfaces.

As a methodological test bed, titanium dioxide has the advantages of relatively simple structure, no magnetic effects, and abundantly available experimental and first principles calculational information. TiO2 is also an important material for engineering applications; it has a very large dielectric constant, and its properties at microwave frequencies are exploited in microwave devices. Many of its close cousins, the perovskite titanates, are ferroelectric.

Stoichiometrically pure titanium dioxide is an insulator, but if, as is usually the case, it is somewhat oxygen deficient, it self-dopes with electrons and is an n-type semiconductor. Because the dielectric constant of the material is very large, the carriers are not believed to propagate like free electrons but are surrounded by a cloud of lattice distortion that increases the effective carrier mass significantly. Such a combination of electron-plus-lattice distortion is called a polaron. Theories of polaron structure and dynamics go back to the 1950s; however, there are very few detailed calculations of the properties and structure of polarons in specific materials. Schelling first fit the tight binding molecular dynamics model of rutile to first principles calculations of the electronic structure of stoichiometrically perfect TiO2 (using, for the first principles calculations, codes developed in the research group headed by Institute Fellow James Chelikowsky, who is Professor of Chemical Engineering and Materials Science at the University of Minnesota). Schelling then used the model to study the structure of polarons in it by adding an extra electron to a structure during a molecular dynamics calculation in which the electronic structure was redetermined self-consistently after each atomic step. These calculations were more detailed than previous polaron calculations and focused on a higher temperature regime in which the atomic motions could be treated as classical. A surprising result was that the electronic wave function associated with the added electron was localized throughout the finite-temperature simulations. The localization, which became less pronounced at lower temperatures, was interpreted as caused by the thermal disorder of the atoms around the carrier.

To understand how titanium dioxide protects titanium metal from corrosion, one needs to model the titanium/titanium dioxide interface. Schelling was able to show that first principles results for the cohesive energy and band structure as function of volume per unit cell of titanium metal could be rather well reproduced with the self-consistent tight binding methods. These calculations were carried out with new "order N" techniques in which the computational effort scales linearly with the number N of atoms in the system.

He placed samples of titanium metal and titanium dioxide in proximity and allowed the atomic positions to relax to equilibrium (at zero temperature). At the same time, he recalculated the electronic structure after each step in which the atoms relaxed along the forces determined from the electronic structure as calculated in the preceding step. A striking result is that Friedel oscillations of the charge in the metal have been induced on the metal side of the interface (as shown in the figure on page 2). These preliminary results on interfaces are exciting because they open the way to study of a variety of technically important Schottky barrier interfaces in microscopic detail.

Recently, Min Zhuang, a research associate in the Halley research group, has been extending these methods to take account of the effects of electronic spin. The basic idea is to take account of the fact that the electronic structure of the constituent ions of a solid or liquid is governed by Hund's rule, which states roughly that, other things being equal, the electronic spin of an ion will be maximized. To systematically take this into account in the self-consistent tight binding model, the on-site energies of the model are caused to depend both on the charge of the ion and on its total spin. Then, both the charges and the spins of the ions are self-consistently determined in the electronic structure calculations. To test the method, Zhuang has made calculations on rutile MnO2, MnF2, and on the spinel structure LiMn2O4. The latter is of interest as a cathode material for advanced lithium polymer batteries. Zhuang has shown that the method predicts the well known antiferromagnetic structure of MnF2, including the correct magnetic moment per ion, within a few percent. In the case of MnO2, he finds spin spiral structures quite similar to tentative reports in the experimental literature on this material. These results suggest the possibility of predicting microscopic magnetic structures using these methods.

Other members of the Halley research group are working on the application of these methods to water, to other transition metal oxides, and to noble metals.